Spectral Analysis for Fourth Order Problem Using Three Different Basis Functions

Transcription

1 Volume 119 o. 1, 09- ISS: 11-9 (on-line version url: ijpam.eu Spectral Analsis for Fourth Order Problem Using Three Different Basis Functions 1 Sagitha and Rajeswari Seshadri 1 Department of Mathematics, Pondicherr Universit, Puducherr, India. sagitha.tm@gmail.com Department of Mathematics, Pondicherr Universit, Puducherr, India. seshadrirajeswari@gmail.com Abstract In this paper, the role of orthogonal functions as a basis functions to obtain a better accurac of solutions of differential equations using spectral methods has been analsed. Three sets of orthogonal functions such as Legendre, Chebshev and Jacobi functions are considered as basis functions for the present stud and their role as a basis function in obtaining a more accurate solution with less computational effort is computed. The stud is carried out for a non homogeneous partial differential equations of fourth order for a set of collocations points. Three functions, either in the form of trigonometric function or a Polnomial form or an exponential form has been taken as non-homogeneous function. For a chosen basis function, for a given non-homogeneous part, the solution is computed for various values of (number of basis functions used in the truncated solution expansion. The error analsis with respect to these three basis functions for the solution of a fourth order PD are discussed. The stud shows that Jacobi polnomial function as basis function gives a faster convergence to an accurate solution compared to Legendre and Chebshev polnomials as basis functions. This analsis in done with the focus to identif and use a suitable basis function for solving stabilit equations in fluid flows which are generall a fourth order differential equation. Ke Words:Spectral solution, fourth order PD, basis function, orthogonal polnomial. 09

2 1. Introduction As is popularl known, several phsical and engineering processes when modelled mathematicall results in partial differential equations. Thus, a search for a more accurate solution of these differential equations using latest developed methods and techniques is a popular research stud in the literature. In the 190 s, there was a revival of Fourier series methods, which were fundamental for a fast calculation of the nonlinear terms through the pseudo spectral technique wherein the differentiations were made in the spectral space. The spectral space represents the space of the expansion coefficients and the products are performed in the phsical space of the values of the unknowns at some discrete points thus establishing the connection between both spaces. Since then, spectral methods have become ver popular and most useful technique especiall for problems in fluid dnamics and turbulent flow analsis. Spectral methods are a class of mathematical techniques to numericall solve certain differential equations. It is the most recent technique presentl used in applied mathematics and in scientific computing to solve ordinar and partial differential equations [1,, 1].The Spectral methods give rise to a class of discretization schemes for solving differential equations and it is a collective name for spatial discretization methods. The essence of spectral methods is that, a set trial functions are used as the basis functions to represent solutions which have truncated series expansion. Thus, it relies on the expansion of solution as coefficients using certain basis functions. The coefficients of this series are unknown which have to be computed. These coefficients are known as spectrum of the solutions, hence the name. Spectral method can be thought of as a generalized version of the finite element method. The advantage of spectral method is that, the basis functions considered are global in nature, which means it can be infinitel differentiable function, whereas in finite element method basis functions are local in nature, (i.e. the domain is divided into small elements and basis function are specified in each element. The set of basis function we choose should be eas to compute and it should converge rapidl and the solution should have high accurac when taking truncation to be large. One of the main advantage of spectral methods is its fast rate of convergence, which is exponential for infinitel differentiable functions. Since we wish to stud the stabilit analsis in laminar flow using spectral methods, as a pilot stud to choose the most optimal basis function to express the solution expansion, the present stud on the choice of basis functions is undertaken. Hence, here we consider the solutions of a fourth order non homogeneous partial differential equation as a series expansion of various basis functions using spectral method. Consider a general differential equation Lu ( x, = f ( x, 09

3 where L ma in general be a linear differential operator. Here we consider L as a partial differential operator. Assuming that the unknown solution function u ( x, can be approximated b a sum of 1 basis functions ( x, such that u( x, = u ( x, = [ a ( x ( ] Then, u ( x, is exact Solution ; u ( x, is approximate solution. Consider the Residual function R as R( x ; ; a = Lu ( x, f ( x, If u( x = u ( x then R 0. ij If not, the challenge is to choose the series coefficients so that the residual function R is minimum. The minimization of Residual function is carried out b choosing a set of +1 collocation points on which the residual R is required to vanish. Different spectral methods such as Galerikan method, Tau method,pseudo spectral methods differ in their minimization strategies. Steven Orszag [,, 11]started a series of research work on spectral methods which resulted in solving Fourier series methods for periodic geometr problems, polnomial spectral methods for finite and unbounded geometr problems, pseudo spectral methods for highl nonlinear problems, and spectral iteration methods for fast solution of stead state problems. Later Gottlieb and Orszag [] summarized the theor and application of spectral methods in their book published in 19. The application of spectral method both in fluid dnamics and meteorolog have been mentioned in the review article of Jarraud and Baede in 19 and also b Haltiner and Williams [, ]. The most popularl used basis function are Jacobi, Legendre, Laguree, Hermite, Chebshev, Trignometric polnomials. Though few studies [, ] suggest the choice of basis functions, there is no hard and fast rule that one basis function gives the best accurac over others for an problem. The choice of basis function depends mainl upon the given equation and its boundar conditions.. Present Stud In this stud, the main focus is to analse the role of orthogonal functions as basis functions to express the solution of fourth order non homogeneous PD as a truncated series of three different basis functions, namel, Chebshev, Jacobi and Legendre functions. As the spectral method suggests, the solution function is first expressed as a linear combination of an of these basis functions. The real coefficients in the series expansions (also called the weights are evaluated 09

4 b a suitable selection of collocation points in the solution domain. The residual function is minimized at these collocation points. The nonhomogeneous term considered here ma be an exponential, a trigonometric or a polnomial functional form. For a given basis function and a non homogeneous term, a set of solution functions u ( x, were derived for several values of where represents the number of polnomial terms considered from the chosen basis function. The computational procedure was performed using all the three basis functions for all combinations of non homogeneous parts and for several values of. A detailed error analsis has been computed for the set of varing from to. Method of Analsis Consider the fourth order PD given b u u = f ( x, xxxx (1 where 1 < x, < 1 and u ( x, satisfies the boundar condition u ( 1, = 0 and u (x, 1 = 0 and u x ( 1, = 0 and u (x, 1 = 0 on the boundar. Here we use three different basis functions (viz., Legendre, Chebshev and Jacobi functions for spectral analsis of the solutions of the fourth order PD considered above. The non-homogeneous part ma be either a trigonometric function or a polnomial function or an exponential function. Solution Using Chebshev Polnomials as Basis Functions ow, using spectral method, with Chebshev polnomials as basis function the approximate solution u can be written as u ( x, = aijti [ x] Tj[ ] ( i=0 j=0 Here T i [x], T j [] are the Chebshev polnomials in x and, respectivel and s a ' ij are the coefficients in the series expansion to be evaluated using a set of collocation points.the residual equation becomes R( x,, aij = u u f ( x, ( xxxx The most suitable form of initial approximation to the solution function is of the form k u = (1 x (1 aijti [ x] Tj[ ] ( k i=0 j=0 where the solution is multiplied with the factor (1 x *(1 in order to 09

5 satisf the boundar conditions.differentiating equation ( four times with respect to x and and substituting in (, we get a residual function. Since the algebraic equation is ver big, it is not given here explicitl. The three non-homogeneous terms in the form of trigonometric, polnomial or exponential functions are taken as follows. 1. f ( x, =*sin x( 1 f ( x, = ( x x * x.. f ( x, = e (( 1 ( x (0. To start with, the solution u ( x, is written as a truncated series expansion using the Chebshev function for a given. Then the residue function is written considering an one of the non homogeneous term f ( x,. For illustration, let us consider =. The number of collocation points required for = are exactl points in the solution domain. Let us take the the non homogeneous part f ( x, as trigonometric function, sa f ( x, =*sin x( 1. In order to compute these unknown coefficients s a ', consider a set of collocation points as i j x ij = ( Cos(, Cos(, where i,j var from 1 to -1. n n Substituting these set of collocation points in residual equation ( and solving the equations we obtain the coefficients a ' to be a 00 = 0.000, a = ,a 01 = 0.00, a 11 = , a 1 = , a 1 = ,a = ,a 0 = 0.000, a 0 = 0.000, a 0 = ,a 1 = 0.000, a 1 = , a = , a =.9* -, a = 0.000,a 0 = 0.000, a 0 = 0.000,a 1 = , a = , a = , a 0 = , a = , a = , a = , a 1 = Thus substituting these values of s a ' ij s ij in the truncated series expansion, we get the approximate solution u ( x, for f ( x, as trigonometric form as u = ( 1. x ( 1. (.*.00* ( = 1.* x ( * x ( * 1.9* * 0.0 x (1.* 1 9.*.* * * 1.9* 9.* 0.1.* x( ij 09

6 Similarl, considering the non homogeneous parts as exponential functional (( 1 ( x (0. form given b f ( x, = e and repeating the above procedure we get the solution as follows: ( = ( 1. x ( 1. ( u = x( x (0.009 x ( x ( and for the polnomial tpe non homogeneous tpe givenb f ( x, = ( x x * x the solution becomes ( = ( 1. x ( 1. ( u = x ( x ( x ( x( The series solution is obtained for different values of varing from to. It is to be mentioned that for =, the number of collocation points required are 19 and in order to compute the 19 coefficients a ', a set of simultaneous equations in 19 unknowns were solved. However, when absolute error k = Uk Uk 1 was computed at ever stage, it was observed that the error 1 was decreasing drasticall and was close to. Solution Using Jacobi Polnomials as Basis Functions In the similar wa, the solutions were obtained for the second set of basis (, (1,1 function namel Jacobi polnomials. we considered P n of the form P n and we choose the collocation points for Jacobi to be the zeroes of derivative of (1,1 P n. The solution for (1 with Jacobi polnomial as basis function and for non homogeneous part as xponential,trigonometric and Polnomial function are given b following equations. For f ( x, as xponential form: s ij u = = 1 ( 1 x ( 1 ( x ( x ( x( x (

7 For f ( x, as Trigonometric form: u = = 1 ( 1 x ( * x( * x (.9* (.0*.* *.9*.9*.99.9* 1 1.9* 0.1.9* x (.9*.0* 1.*.0* x ( For f ( x, as Polnomial form: 1 u = = ( 1 x ( 1 ( x ( Solution Using Legendre Polnomials as Basis Functions 0.19 x( x ( x ( In the same wa, the solutions were computed b changing the basis functions to Legendre polnomials L. The Collocation points are given b zeroes of n derivative of L n. Again for the value of =, the solutions for the three non homogeneous cases are given as follows. For f ( x, as xponential form: u = = ( x( 0.09 x ( x ( x ( x ( 1. ( For f ( x, as Trigonometric form: u = = 0.009( x ( For f ( x, as polnomial form u = = 0.9( 1. x ( x ( x ( x( x ( 1. ( x ( (0.0 x( x ( x (

8 . Results and Discussion The solution of the fourth order PD has been computed for various forms of non homogeneous functions using three different basis functions. In all the cases, the computation is performed using Mathematica.1 software. The number of basis functions ( used in the solution expansion var from = to =. Hence, for each case, the collocation points at which the residual function is evaluated varies from to 19. With the choice of three basis functions, three non homogeneous functional forms and 1 sets of values ( varing from to, the total number of solutions derived were x x 1 = 1. There were 1 u ( x, expressions derived. As becomes larger, the expressions get too big. The functional values of each of these 1 analtical expressions were evaluated at several points of the solution domain. Here for our calculation, we have allowed both x and to var from -1 to 1 in steps of That is to mention that each u ( x, was evaluated at approximatel 00 points in the domain [-1, 1] x [-1, 1].The results obtained are presented in the form of some representative tables. Several runs have been made for various s and changing basis functions and non homogeneous parts. Tables 1 to present the absolute error of the solution function for the three tpes of non homogeneous parts, namel exponential, trigonometric and polnomial function, with respect to all the three basis functions. The table value shows the maximum error that can occur at an point in the solution domain [ 1,1] x [ 1,1] and is found to be of the order, and for exponential, trigonometric and polnomial non homogeneous parts, respectivel. Table 1: Comparison of rror between the Three Basis Function with xponential on Homogeneous Part rror Chebshev Legendre Jacobi. *. *.019 *.01 * 1. *.00 * 1 1. *.9 *. * 1.1 *.91 *.9 *. * 1.9 * 1.9 * 1.9 *.0011 *. * 0

9 Table : Comparison of error between the three basis function with Trigonometric non homogeneous part rror Chebshev Legendre Jacobi.09 *. *.9 * 1.9 *.91 *.9 *. * 1.11 *.90 *.901 * 1.19 *. * 1. *. *.90 * 1.9 *. * 1. * Table : Comparison of error between the three basis function with Polnomial non homogeneous part rror Chebshev Legendre Jacobi * 1. * 1.0 * 1 1. *. * *.10 * 1. *. *. *.1 * 1. *. * 1.09 * *. *.1 * 1.9 * Tables to give the absolute errors of the solution at eight arbitrar coordinates in [ 1,1] x [ 1,1] domain with regard to the three basis functions for the nonhomogeneous part as exponential,trigonometric and polnomial functions. Table presents the range of error committed in evaluating the solution of Fourth order PD using the three different basis functions and three different non homogeneous parts. It is observed that the maximum error at an point within the solution domain is for an form of non homogeneous part, irrespective of whatever basis function we choose. However, if the non homogeneous part is a polnomial function, sa, then the preferred basis function is Jacobi functions as it converges much faster and the 1 maximum error is and the minimum error is which is close to zero. Table : The absolute error of the solution at different coordinates with respect to the three basis function and with xponential function as non homogeneous part ( x, ( 0.,0. ( 0., 0. (0.,0. (0,0. Jacobi 1 9.9*. * 1.99* 9. * Legendre *. * 9. * 1.90 * Chebshev 9.9*.19 * 9.9 *.1 * ( x, (0.,0. (0.,0. ( 0., 0. (0, 0. Jacobi * 9. * 1 1.*.* Legendre 9 1. * 1.90 *.* 1. * Chebshev 9.9 * 1.9 * * 11. * 1

10 Table : The absolute error of the solution at different coordinates with respect to the three basis function and with Trigonometric function as non homogeneous part ( x, ( 0.,0. ( 0., 0. (0.,0. (0,0. Jacobi. *.9 *.91 *.91 * Legendre 9.1 *.0 *.1 *.9 * Chebshev 1.0 *.1 *.0 * * ( x, (0.,0. (0.,0. ( 0., 0. (0, 0. Jacobi. * *.09 * 1.1 * Legendre 9.1 *.1 *. *. * Chebshev 1.0 *.9 *. *. * Table : The absolute error of the solution at different coordinates with respect to the three basis function and with Polnomial function as non homogeneous part ( x, ( 0.,0. ( 0., 0. (0.,0. (0,0. Jacobi 1.0 *.91 *.*.99* Legendre.0*.99 *.191 * 1.0 * Chebshev.9*.99 *. *.0* ( x, (0.,0. (0.,0. ( 0., 0. (0, 0. Jacobi.9 * 1.09 * * 1.0 * Legendre.19*. *.* 1. * Chebshev.*. * 1.9 * * Table : Range of rror in the domain [-1,1] x [-1,1] on Homogeneous PartBasis Function Maximum rror Minimum rror xponential Chebshev.9 *.* Legendre.90 * 1. * Jacobi 1. * 0 Trigonometric Chebshev.1 *.900 * Legendre 1.0 *. * Jacobi 1.19 *. * Polnomial Chebshev 9.0 *.0* Legendre.009 * 1.1 * Jacobi 1. * 0. Conclusion Three orthogonal polnomials such as Legendre, Chebshev and Jacobi functions are considered as basis functions to obtain the solution of a fourth order non homogeneous PD using Spectral method. The non homogeneous term ma be an exponential tpe or a trigonometric form or a polnomial function. For each non homogeneous case, we have expressed the solutions with

11 respect to all the three basis functions. For each case, as we increase the number of orthogonal polnomial terms from to, we computed how the solution improves b calculating the absolute error at each stage. The detailed error analsis is done which shows that, the maximum error that can occur at an point in the solution domain [ 1,1] x [ 1,1] is where as the minimum 1 error is of the order of for. The authors intend to stud the stabilit analsis in fluid flow using spectral methods which is a fourth order nonlinear differential equation leading to an eigen value problem. Thus, the present stud was carried out to understand suitabilit of a basis function to carr out the stabilit analsis in obtaining a more accurate solution with less computational effort. Hence, with this pilot stud, it was observed that, for the fourth order PD considered, among the three basis functions, the Jacobi polnomials as basis functions show a faster convergence compared to the other two basis functions for the choice of non homogeneous parts considered here. However, we wish to conclude that, the choice of the basis function leading to faster convergence depends upon the nature of the differential equations as well the boundar conditions. Hence the present stud is ver useful in identifing a suitable set of orthogonal polnomials as basis functions in order to stud phsicall realistic fluid flow problems. Acknowledgment The first author wishes to acknowledge the financial support received from UGC-BSR Research Fellowship, ew Delhi, Government of India. References [1] Bengt Fornberg.(199.A Practical Guide to Pseudospectral Methods. Cambridge Universit Press, Cambridge. [] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang.(1991.Spectral Methods in Fluid Dnamics. Springer-Verlag, ew York. [] D. Gottlieb and S. A. Orszag.(19. umerical Analsis of Spectral Methods. SIAM, Philadelphia [] George J. Haltiner and Roger T. Williams.(190.umerical Prediction and Dnamic Meteorolog. Wi-le. [] Jarraud, M. and Baede, A. P. M.(19. "The Use of Spectral Techniques in umerical Weather Pre-diction," in Large-scale Computations in Fluid Mechanics. Lectures in Applied Mathematics,, 1-1. j

12 [] J.P. Bod.(001.Chebshev and Fourier spectral methods. nd edition, Dover, Mineola. [] J.P. Bod, R. Petschek.(01. The relationships between Chebshev, Legendre and Jacobi polnomial: The generic superiorit of Chebshev polnomials and three important exceptions.j. Sci. Com-put., 9(1, 1-. [] Karniadakis, G.. and Orszag, S. A.(199. Some novel aspects of spectral methods, in M. Y. Hus-saini, A. Kumar and M. D. Salas (eds.algorithmic Trends in Computational Fluid Dnamics,ICAS/ASA LaRC Series, Springer-Verlag, ew York, -. [9] Llod. Trefethen.(000.Spectral Methods in MATLAB. SIAM, Philadelphia, PA. [] McCror, R. L. and Orszag, S. A.(190. Spectral methods for multi-dimensional diffusion problems.journal of Computational Phsics, (1, [11] Orszag, S. A.(19. Comparison of pseudospectral and spectral approximations. Studies in AppliedMathematics,1(, -9. [1] W.Huang and David.M.Sloan.(199. The Pseudospectral method for Third order differential equations. SIAM. J.umer.Anal.,9(, -.

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